document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=960751>Question 960751</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The radiator of a truck contains 12 gallons of water. We draw off 3 gallons and replace it by alcohol, then draw off 3 gallons of the mixture and replace it by alcohol,etc., until 8 drawings and replacement have been made. How much alcohol is in the final mixture? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #779587 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=779587\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The analysis by the other tutor is faulty....<br><BR>\n" );
document.write( "And if you evaluate the expression he shows for the amount of alcohol after the 8 drawings and replacements, it is more than 12 gallons -- which of course is nonsense.<br><BR>\n" );
document.write( "I did some ugly calculations for the first few drawings and replacements and noticed a clear pattern in the results; then I verified my calculations using an excel spreadsheet.<br><BR>\n" );
document.write( "Then I finally saw the simple explanation of the pattern I saw, and thus an easy path to the answer to the problem.<br><BR>\n" );
document.write( "Each time a drawing and replacement is made, 1/4 of the mixture is taken out and replaced with pure alcohol.  That means that with each drawing and replacement, the amount of water remaining gets reduced by a factor of 3/4.<br><BR>\n" );
document.write( "So after 8 drawings and replacements, the amount of water remaining is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%283%2F4%29%5E8 ALIGN=MIDDLE  ALT=\"12%283%2F4%29%5E8\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 1.201355 gallons, to far more decimal places than practical.<br><BR>\n" );
document.write( "So the amount of alcohol in the radiator at that time is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12-1.201355 ALIGN=MIDDLE  ALT=\"12-1.201355\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 10.798645 gallons<br><BR>\n" );
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